Problem - 2422
Solve the following system in integers:
$$
\left\{
\begin{array}{ll}
x_1 + x_2 + \cdots + x_n &= n \\
x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\
\cdots\\
x_1^n + x_2^n + \cdots + x_n^n &= n
\end{array}
\right.
$$
Investigate the function $f(x)=(x-x_1)(x-x_2)\cdots(x-x_n) = \displaystyle\sum_{k=0}^n a_k x^k$.
Obviously, $f(x_1)=f(x_2)=\cdots=f(x_n)=0$. Therefore
$$0=\sum_{k=1}^n f(x_k) = \sum_{j=0}^n \Big(a_j\sum_{k=1}^n x_k\Big)= n \sum_{j=0}^n a_j $$
Hence we have $$f(1) = \sum_{j=0}^n a_j = 0 \implies 1\text{ is a root}$$
Or one of $x_i$ equals 1.
Repeat this process can lead to all $x_i$ equal 1.